1. Field of the Invention
The present invention generally relates to electronic circuits and, more specifically, to integrated circuits comprising calculation elements (software and/or hardware) implementing algorithms performing several identical operations on a same element of a group in the mathematical meaning of the word. “Operation” is used to designate any law of composition of two elements in the group such that the result is an element in the group. The term “operation” and the expression “internal (composition) law” will be used interchangeably hereafter.
An example of application of the present invention relates to exponentiations especially performed in cryptographic calculations, for example of a so-called RSA algorithm which is a public key algorithm based on a modular exponentiation calculation.
Another example of application of the present invention relates to operations performed on elliptic curves in cryptographic calculations, for example, of a so-called Diffie-Hellman algorithm which is a protocol for generating shared keys from an exchange of public keys.
The present invention more specifically relates to the protection of an iterative calculation against attacks by disturbance of the operation of the electronic circuit executing the calculation. Such attacks for example aim at discovering quantities intended to remain secret. The present invention more specifically relates to so-called fault-injection attacks.
An example of application of the present invention relates to smart cards and the like.
2. Discussion of the Related Art
FIG. 1 very schematically shows a smart card 1 of the type to which the present invention applies as an example. Such a card is most often formed of a plastic support on or in which is placed an integrated circuit chip 10 associated with contacts 5 of communication with a terminal (not shown) and/or with ratio-frequency transmit/receive elements, not shown, for a contactless communication.
Another example of application of the present invention relates to microcomputers and more generally electronic boards (for example, a personal computer motherboard) comprising an integrated circuit performing calculations on a group, for example, for data transmission.
FIG. 2 very schematically shows, in the form of blocks, a conventional example of electronic circuit 10, for example, of a smart card, to which the present invention more specifically applies. In this example, it is a microcontroller comprising a central processing unit 11 (CPU) capable of executing programs most often contained in a ROM 12 with which unit 11 communicates by means of one or several buses 13. Bus(es) 13 convey signals between the different circuit elements and especially between central processing unit 11 and one or several RAMs 14 intended to contain data being processed, and an interface 15 (I/O) for communicating with or without contact with the outside of circuit 10. In circuits to which the present invention applies, a ciphering or cryptography function 16 (CRYPTO) is most often implemented in hardware fashion in microcontroller 10 and executes at least one calculation, for example, of exponentiation. The microcontroller may also comprise a rewritable non-volatile memory area 17 (NVM) (for example, of EEPROM type or the like) and other functions (block 18, FCT) according to the application, for example, cyclic redundancy check (CRC) functions, functions of generation of digital quantities for a DES-type algorithm, etc.
The present invention also applies to simpler integrated circuits only having a cryptographic calculation unit and a memory area for storing at least one or several quantities intended to remain secret and defining the number of iterations of the operation.
FIG. 3 very schematically illustrates in the form of blocks a conventional example of an RSA algorithm exploiting a modular exponentiation calculation. Such an algorithm is described, for example, in the “Handbook of Applied Cryptography” by A. Menezes, P. Van Oorschot, and S. Vanstone, published by CRC Press in 1997, and in RSA Cryptography Standard V2.1 (RSA Labs, Jun. 14, 2002), which is incorporated herein by reference.
A message M to be ciphered is sent to a ciphering cell 161 (for example, a dedicated portion of an integrated circuit) which also receives or contains an exponent e and a modulo n to be used in the calculation and which define the public key of the RSA algorithm. Block 161 executes calculation Me mod n and provides ciphered message M′.
On the deciphering side, a modular exponentiation cell 162 receives message M′, as well as modulo n (public in the RSA case) and an exponent d here defining an element of the private key of the message receiver. The performed calculation is identical to that of the ciphering. Cell 162 executes operation Md mod n to provide message M plain. The possible relations that numbers e, d, and n should respect for the implementation of the RSA algorithm are of no importance for the discussion of the present invention. In practice, the same circuit may comprise a single cell 161 or 162 loaded with different parameters according to whether it ciphers or deciphers.
Due to the size of the handled numbers, the exponentiation is calculated by a so-called square-and-multiply technique which exploits the binary representation of the exponent (e or d) to break up the calculation into a succession of squarings and multiplications by a preceding intermediary result.
FIG. 4 is a flowchart illustrating an exponentiation calculation by a conventional square-and-multiply technique. The calculation, shown in the form of a flowchart in FIG. 4, is in practice generally performed by a hardware cell (in wired logic) but may also be implemented by software means.
A first step (block 21, R=1; T=M; e′=e) comprises initializing a result variable R to one, a temporary variable T as containing message M, and an exponent variable e′ to the value of exponent e. In the RSA case, all calculations are performed modulo n. Value n is thus also known or received by the cell for executing the exponentiation.
To simplify the discussion of the present invention, an exponentiation calculation will, for example, be taken with notations Me mod n, knowing that number M, exponent e, and modulo n may form all or part of any number (for example, M′), exponent (for example, d), and modulo, in relation or not with the RSA algorithm.
The square-and-multiply technique takes advantage of the binary expression of the exponent in a calculation by electronic or computer means. Variable e′ will be considered hereafter as a succession of bits initially representing exponent e of the calculation.
The square-and-multiply technique is performed by iterations on variables
and R, the number of iterations being equal to the number of (significant) bits of exponent e.
Before each iteration, the current value of variable e′ is tested (block 22, e′=0?) to determine whether it still contains significant bits (at least another bit at 1). If variable e′ is zero (output Y of test 22), result variable R provides result Me of the exponentiation. Otherwise (output N of block 22), the calculation enters a loop.
At each iteration of this loop, the even or odd character of the current value of variable e′ is tested (block 23, Is e′ ODD ?). If e′ is odd (output Y of test 23), the content of variable R is multiplied by the content of variable T and the result becomes the current value of variable R (block 24, R=R*T). Otherwise (output N of block 23), variable R is not modified.
The content of variable e′ is then shifted rightwards (block 25, Right SHIFT e′), which amounts to eliminating the least significant bit which has conditioned the even or odd character in the preceding test 23. In the example of a binary representation of the exponent, this amounts to dividing variable e′ by 2 (in integer part). According to the hardware elements used to execute the algorithm, the step of rightward shifting of variable e′ may be carried out by a shift register or be replaced with the successive taking into account of the different bits of exponent e.
The content of variable T is then squared (block 27, T=T*T), which amounts of performing another multiplication and the result becomes the current value of variable T. The iteration is over. The calculation then resumes with test 22 to restart an iteration if there remain unprocessed significant bits of the exponent.
Optionally, the current value (comprising one less representative bit) of variable e′ is tested (block 26, e′=0 ? in dotted lines) prior to step 27 to check whether there remain significant bits. If so (output N of test 26), calculation 27 is performed. Otherwise (output Y of block 26), variable T is not modified. This option enables saving a calculation at the end of the algorithm.
A disadvantage of an exponentiation calculation such as described in relation with FIG. 4 is that it is vulnerable to attacks tending to discover the handled secret quantities.
A first type of attack is to monitor the calculation execution time which differs, at each iteration, according to the even or odd character of current exponent e′. Now, this even or odd character directly provides value 0 or 1 of the corresponding least significant bit.
A known solution to solve this execution time problem is to introduce (block 28, D=R*T), in case of a negative test 23 (least significant bit of variable e′ equal to 0), an arbitrary calculation that is not needed for the result.
Such an unnecessary calculation is not necessary to compensate for calculation 27 in the presence of test 26, since the only iteration in which calculation 27 is not executed is the last one.
Even if it can be provided for the calculation to take the same time whatever the iteration, this calculation remains vulnerable to another category of attacks, called fault injection attacks. A fault injection attack comprises causing a disturbance in the integrated circuit operation in the calculation execution (for example, by means of a laser, of a disturbance on the power supply, etc.) and interpreting the subsequent circuit operation to attempt discovering the secret quantities (here exponent e).
In the case of a square-and-multiply calculation, by disturbing the multiplication operation (block 24 or block 28) in an iteration of the algorithm, a hacker is able to determine whether the least significant bit of the current value of the exponent (e′) is 0 or 1. Indeed, if it is a 0, the final result provided by the calculation will be the same as with no disturbance, since the disturbance bears on the multiplication of block 28 while, if the bit is at 1, the final result will be modified since the disturbance bears on the multiplication of block 24. By repeating the fault injections at different times in successive executions of the same calculation, it is then possible for the hacker to deduce all or part of the key (the exponent), and thus, at least, to decrease the number of assumptions to be made about this key.
FIG. 5 very schematically illustrates in the form of blocks another example of application of the present invention to a Diffie-Hellman algorithm on an elliptic curve.
Such an algorithm is used to create a shared key K (for example, a session key) for a protected exchange between two systems or circuits P1 and P2. Each circuit holds a private key, respectively a or b, and an element G of an elliptic curve defining, with an internal composition law arbitrarily called “addition”, noted “(+)”, an abelian group on which the calculation is performed. Element G is known and needs no protection. Private keys a and b are integers expressing the number of times that the group composition law is applied.
A property of an elliptic curve thus is that, starting from a point in the curve, the application, an integral number of times, of the composition law called addition provides a result still located on the curve. Sometimes, it is spoken of as a “product”, noted “.”, to designate the number of times that the composition law is applied.
Elliptic curves are used in cryptography for the asymmetrical character of the iterative calculation, that is, knowing a point in the curve, it is easy to obtain another point in the curve by applying the composition law an integral number of times but, knowing two points in the curve, it is difficult to find the integer (the key) connecting these two points.
Examples of application of elliptic curves to cryptography are described in standards: “Standards for Efficient cryptography, sec 1: Elliptic Curve Cryptography”—Certicom Research—Sep. 20, 2000, Version 1.0c; “DSA on Elliptic Curves: ECDSA”—ISO/IEC 15946-2; and “Diffie-Hellman on Elliptic Curves: ECDH”—ISO/IEC 15946-3, which references are incorporated herein by reference.
Each circuit P1, P2 calculates a public key, respectively A, B as being the result of the “product” of its private key a or b with element G (block 163, A=a.G and block 165, B=b.G). Then, each circuit sends its public key to the other. Finally, each circuit P1, P2 calculates a key K as being the “product” of its private key a or b by the public key, respectively B or A, of the other circuit (block 164, K=a.B, block 166, K=b.A). Due to the properties of the internal composition law of the group, keys K are identical (K=a.b.G=b.a.G). Key K can thus be used as a secret key shared by the two circuits. It can then be used as a ciphering (for example, by a symmetrical algorithm), authentication, etc. key between circuits P1 and P2.
As for an exponentiation, due to the size of the handled numbers, the successive applications of the internal composition law from a point of an elliptic curve are calculated by a so-called add and double technique which exploits the binary (polynomial) representation of the integer (a or b) to break up the calculation into a succession of doublings (application of the composition law to a point on itself) and additions (application of the composition law to a preceding intermediary result).
FIG. 6 is a flowchart illustrating the add and double technique, for example, applied to a point G of an elliptic curve. As for FIG. 4, the calculation shown as a flowchart in FIG. 6 is in practice generally performed by a hardware cell (in wire logic) but may also be implemented in software fashion.
The add and double technique exploits the same characteristic of the binary expression of the integer as that exploited for the exponent in the modular exponentiation. Indeed, product a.G can be written as G(+)G(+) . . . (+)G (a times). Representing a in binary fashion over t bits, this product can also be written as at2tG(+)at−12t−1G(+) (+)a121G(+)a020G, where coefficient ai of the term of degree i (i ranging between 0 and t) corresponds to the bit (value 0 or 1) of weight i of the binary expression of number a.
The add and double calculation is then performed by iterations on variables T and R, respectively representing a temporary variable and the result variable, the number of iterations being equal to number t of (significant) bits of number a. The calculation of FIG. 6 is identical to that of FIG. 4, by:
initializing (block 21′, R=0; T=G; a′=a) variables R, T, and a′, respectively with values 0 designating the neutral element of the internal composition law (here, the addition) on the group (here, the elliptic curve), G designating the group element (here, a point of the elliptic curve) and a in the form of a sequence of coefficients of a polynomial (here, a bit sequence), the significant degree of which designates the number of iterations;
replacing the multiplications * of blocks 24 and 27 (and, if existing, that of block 28) with the (+) composition law (block 24′, R=R(+)T, block 27′, T=T(+)T and block 28′, D=R(+)T); and
applying steps 22, 23, and 25 to variable a′.
The algorithm of FIG. 6 exhibits the same disadvantages of vulnerability to attacks, especially by fault injections, as the square-and-multiply algorithm.
More generally, similar problems may be posed for any calculation, by an electronic circuit, of an integral number of applications of an internal composition law on elements of an abelian group, by successive iterations of different steps according to the even or odd character of a current coefficient of a polynomial representation of said integer, the degree of which determines the number of iterations.
In the case of the exponentiation, the exponent is the integer and the composition law or internal operation of the abelian group is multiplication.